Those are 10SQ050 50 V 10A Schottky diodes which are specifically sold as solar panel "anti shading" bypass diodes. They should not be removed - removing them produces absolutely no change during normal operation but removes a useful function when the panel is shaded.
From your description it is possible that 2 of the diodes are used in shading bypass mode and one is in series with the panel output. If this is the case then the output would be reduced by an extremely small amount in operation but this would have zero affect on operation in most cases and utterly minimal effect if a high quality MPPT converter was used. These are very good performance diodes - series voltage drop at rated current is lower than most Schottky diodes usually encountered - probably around 0.6V at full current. If the panel is used with a non-MPPT converter this voltage drop will have no affect on the available power. If used with an MPPT converter it would perhaps reduce output by about 1% to 2% in full sun. Other variable losses would swamp this - non solar tracking, dirt on panel, temperature effects etc.
Without a series diode SOME battery energy will drain through the panel. This will usually be nowhere near as large as full panel current as a non illuminated panel acts as a low current current sink when voltage is applied. Some greater losses may occur during the dawn and dusk periods as panel voltage rises but is below battery voltage.
They do no harm whatsoever in normal use, do not reduce panel output in any way and allow the panel to produce some output when cells in one of the series strings are shaded.
In a high voltage system with multiple panels they also provide protection against reverse voltage destructive breakdown if the panel is shaded while the rest of the array is in bright sun. If a cell breaks down in such conditions it is common for a physical hole to be burned in the panel where the shaded cell used to be. [When only a single panel is used the voltages experienced are extremely unlikely to be high enough to cause this effect].
Yes, you can remove the diodes. But you need to be aware that the solar panel will act as a load itself when the amount of light drops below a certain amount.
If the panel were connected to a battery bank and the charge controller doesn't prevent reverse current flow, the solar panel will discharge the battery.
Upon closer look at your photo, I think that those diodes are intended to allow the solar panel to continue to function should a portion of the panel be shaded.
I'm guessing that there are 3 major strings of solar cells all connected in series. Note that the diodes are connected across each of those strings.
If one of those strings of cells becomes shaded, the current available from that string will drop to a lower value than the other strings of cells in the panel. This will cause the voltage across that particular sting to go negative and thus reduce the amount of power available from the panel.
The diode connected across the shaded string will clamp the reverse voltage to about one diode drop. The diode will then pass the current from the remaining strings to the output leads of the panel.
Those look like high-current bypass diodes used in case of an array section going bad or if a local fuse blows. The diodes allow any remaining good panels to keep supplying power, even if there is a 33% drop in voltage. They allow for up to 3 panels to be in series to build up higher voltages.
Regulator / Charger IC's handle these issues to keep battery charging as stable as possible during daylight hours.
You can never take the diodes out, because if one panel failed or blew a fuse, the power from the solar panel would drop to zero volts.